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Guidance

Two right triangles with one pair of non-right congruent angles are similar by
. This means the ratio between the side lengths of the first triangle must be congruent to the ratio between the corresponding side lengths of the second triangle.

For example, in the picture above,
. Because there are
three
pairs
of sides for any triangle, there are
three
relevant
ratios
for a given angle.

The
tangent
of an angle gives the ratio
.
The abbreviation for tangent is tan.

The
sine
of an angle gives the ratio
.
The abbreviation for sine is sin.

The
cosine
of an angle gives the ratio
.
The abbreviation for cosine is cos.

These are the basic
trigonometric
ratios
.
Trigonometry
is the study of triangles. These ratios are called trigonometric ratios because they apply to triangles. Just as your scientific or graphing calculator has
tangent
programmed into it, it also has
sine
and
cosine
programmed into it. This means that you can use your calculator to determine the ratio between the lengths of any pair of sides for any angle within a right triangle.

Example A

Use your calculator to find the sine ratio and cosine ratio for a
angle.

Solution:
and
.

Example B

Solve for
.

Solution:
Look to see how the sides that are marked are related to the
angle. The side of length 11 is
adjacent
to the angle. The side of length
is the
hypotenuse
of the triangle. When working with the adjacent side and the hypotenuse, you should use the cosine ratio.

Note that
is the exact answer. 12.346 is an approximate answer because you rounded the value of
.

Example C

Find
and
.

Solution:
Relative to angle
, 3 is the opposite leg and 4 is the adjacent leg. In order to find the sine and cosine ratios you also need to know the hypotenuse of the triangle. Use the Pythagorean Theorem to find the hypotenuse:

Now, write the sine and cosine ratios:

Concept Problem Revisited

SOH-CAH-TOA is a mnemonic that many people use to remember the difference between sine, cosine, and tangent. How can remembering SOH-CAH-TOA help you?

Vocabulary

Two figures are
similar
if a similarity transformation will carry one figure to the other.
Similar figures
will always have corresponding angles congruent and corresponding sides proportional.

AA, or Angle-Angle
, is a criterion for triangle similarity. The AA criterion for triangle similarity states that if two triangles have two pairs of congruent angles, then the triangles are similar.

The
tangent (tan)
of an angle within a right triangle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

The
sine (sin)
of an angle within a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

The
cosine (cos)
of an angle within a right triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

The
trigonometric ratios
are sine, cosine, and tangent.

Trigonometry
is the study of triangles.

, or
“theta”,
is a Greek letter. In geometry, it is often used as a variable to represent an unknown angle measure.

Guided Practice

1. Use your calculator to find the sine and cosine ratios for a
angle.

2. Solve for
.

3. Find
and
.

Answers:

1.
and
.

2. Look to see how the sides that are marked are related to the
angle. The side of length 17 is
opposite
from the angle. The side of length
is the
hypotenuse
of the triangle. When working with the opposite side and the hypotenuse, you should use the sine ratio.

3. Relative to angle
, 4 is the opposite leg and 2 is the adjacent leg. In order to find the sine and cosine ratios you also need to know the hypotenuse of the triangle. Use the Pythagorean Theorem to find the hypotenuse:

Now, write the sine and cosine ratios:

Note that in the last step of each calculation the denominator was rationalized. You can choose to rationalize the denominator if you wish.

Practice

For #1-#6, use the triangle below. Find each exact value.

1.

2.

3.

4.

5.

6.

Identify whether the sine, cosine, or tangent ratio is most useful for helping to solve the problem. Then, solve for
.

7.

8.

9.

10.

11.

12.

Use the triangle below for #13-#15.

13. Find
.

14. Draw an altitude from
to divide the triangle into two right triangles. Use trigonometry to find the lengths of the sides of each of these right triangles.

15. Find the perimeter of
.

16. True or false: You need to know the length of at least one side of a triangle to find the lengths of the other sides.